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Advances in Difference Equations

, 2018:369 | Cite as

Coupled Hilfer fractional differential systems with random effects

  • Saïd Abbas
  • Mouffak Benchohra
  • Yong Zhou
Open Access
Research
  • 46 Downloads

Abstract

This paper deals with some existence results for two classes of coupled systems of Hilfer and Hilfer–Hadamard random fractional differential equations. The main tool used to carry out our results is Itoh’s random fixed point theorem.

Keywords

Functional differential equation Left-sided mixed Riemann–Liouville integral of fractional order Left-sided mixed Hadamard integral of fractional order Hilfer fractional derivative Hilfer–Hadamard fractional derivative Coupled system Random solution Fixed point 

MSC

26A33 45D05 45G05 45M10 

1 Introduction

Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering, and other applied sciences [13, 23]. For some fundamental results in the theory of fractional calculus and fractional differential equations we refer the reader to the monographs of Abbas et al. [5, 6], Samko et al. [22], and Kilbas et al. [18], and a series of papers [2, 3, 4, 7, 15, 26, 27, 29, 30, 31, 32, 33] and the references cited therein. Coupled systems of Hadamard type sequential fractional differential equations were considered in [8, 9]. Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative; see [1, 11, 12, 13, 16, 24, 28].

The nature of a dynamic system in engineering or natural sciences depends on the accuracy of the information we have concerning the parameters that describe that system. If the knowledge about a dynamic system is precise then a deterministic dynamical system arises. Unfortunately in most cases the available data for the description and evaluation of parameters of a dynamic system are inaccurate, imprecise or confusing. In other words, evaluation of parameters of a dynamical system is not without uncertainties. When our knowledge about the parameters of a dynamic system are of statistical nature, that is, the information is probabilistic, the common approach in mathematical modeling of such systems is the use of random differential equations or stochastic differential equations. Random differential equations, as natural extensions of deterministic ones, arise in many applications and have been investigated by many mathematicians. We refer the reader to Refs. [10, 19, 25].

In this paper we discuss the existence of solutions for the following coupled random Hilfer fractional differential system:
$$ \textstyle\begin{cases} ((D_{0}^{\alpha_{1},\beta_{1}}u_{1})(t,w),(D_{0}^{\alpha_{2},\beta _{2}}u_{2})(t,w)) \\ \quad =(f_{1}(t,u_{1}(t,w),u_{2}(t,w),w),f_{2}(t,u_{1}(t,w),u_{2}(t,w),w));\quad t\in I, \\ ((I_{0}^{1-\gamma_{1}}u_{1})(t,w),(I_{0}^{1-\gamma _{2}}u_{2})(t,w))|_{t=0}=(\phi _{1}(w),\phi_{2}(w)), \end{cases}\displaystyle \quad w\in \Omega, $$
(1)
where \(I:=[0,T]\), \(T>0\), \(\alpha_{i}\in(0,1)\), \(\beta_{i}\in[0,1]\), \(\gamma _{i}=\alpha _{i}+\beta_{i}-\alpha_{i}\beta_{i}\); \(i=1,2\); \((\Omega,\mathcal{A})\) is a measurable space, \(\phi_{i}:\Omega\to {\Bbb {R}}\) is a measurable function, \(f_{i}:I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\to{\Bbb {R}}\) is a given function, \(I_{0}^{1-\gamma_{i}}\) is the left-sided mixed Riemann–Liouville integral of order \(1-\gamma_{i}\), and \(D_{0}^{\alpha_{i},\beta_{i}}\) is the Hilfer fractional derivative of order \(\alpha_{i}\) and type \(\beta_{i}\).
Next, we consider the following coupled system of random Hilfer–Hadamard fractional differential equations:
$$ \textstyle\begin{cases} (({}^{H}D_{1}^{\alpha_{1},\beta_{1}}u_{1})(t,w),({}^{H}D_{1}^{\alpha_{2},\beta _{2}}u_{2})(t,w)) \\ \quad =(g_{1}(t,u_{1}(t,w),u_{2}(t,w),w),g_{2}(t,u_{1}(t,w),u_{2}(t,w),w));\quad t\in[1,T], \\ (({}^{H}I_{1}^{1-\gamma_{1}}u_{1})(t,w),({}^{H}I_{1}^{1-\gamma _{2}}u_{2})(t,w))|_{t=1}=(\psi_{1}(w),\psi_{2}(w)), \end{cases}\displaystyle \quad w\in \Omega, $$
(2)
where \(T>1\), \(\alpha_{i}\in(0,1)\), \(\beta_{i}\in[0,1]\), \(\gamma_{i}=\alpha _{i}+\beta _{i}-\alpha_{i}\beta_{i}\), \(\psi_{i}:\Omega\to{\Bbb {R}}\); \(i=1,2\) is a measurable function, \(g_{i}:[1,T]\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\to{\Bbb {R}}\) is a given function, \({}^{H}I_{1}^{1-\gamma_{i}}\) is the left-sided mixed Hadamard integral of order \(1-\gamma_{i}\), and \({}^{H}D_{1}^{\alpha_{i},\beta_{i}}\) is the Hilfer–Hadamard fractional derivative of order \(\alpha_{i}\) and type \(\beta_{i}\).

The present paper initiates the study of coupled systems for Hilfer fractional differential equations with random effects.

2 Preliminaries

Let C be the Banach space of all continuous functions v from I into \({\Bbb {R}}\) with the supremum (uniform) norm
$$\|v\|_{\infty}:= \sup_{t\in I} \bigl\vert v(t) \bigr\vert . $$
As usual, \(AC(I)\) denotes the space of absolutely continuous functions from I into \({\Bbb {R}}\). We denote by \(AC^{1}(I)\) the space defined by
$$AC^{1}(I):= \biggl\{ w:I\to{\Bbb {R}}:\frac{d}{dt}w(t)\in AC(I) \biggr\} . $$
By \(L^{1}(I)\), we denote the space of Lebesgue-integrable functions \(v:I\rightarrow{\Bbb {R}}\) with the norm
$$\|v\|_{1}= \int_{0}^{T} \bigl\vert v(t) \bigr\vert \,dt. $$
Let \(L^{\infty}(I)\) be the Banach space of measurable functions \(u:I\to {\Bbb {R}}\) which are essentially bounded, equipped with the norm
$$\|u\|_{L^{\infty}}=\inf \bigl\{ c>0: \bigl\vert u(t) \bigr\vert \leq c, \mbox{a.e. }t\in I \bigr\} . $$
By \(C_{\gamma}(I)\) and \(C^{1}_{\gamma}(I)\), we denote the weighted spaces of continuous functions defined by
$$C_{\gamma}(I)= \bigl\{ w:(0,T]\to{\Bbb {R}}: t^{1-\gamma}w(t)\in C \bigr\} , $$
with the norm
$$\|w\|_{C_{\gamma}}:= \sup_{t\in I} \bigl\vert t^{1-\gamma}w(t) \bigr\vert , $$
and with
$$C^{1}_{\gamma}(I)= \biggl\{ w\in C: \frac{dw}{dt}\in C_{\gamma} \biggr\} , $$
with the norm
$$\|w\|_{C^{1}_{\gamma}}:=\|w\|_{\infty}+\|w\|_{C_{\gamma}}. $$
Also, by \({\mathcal {C}}:=C_{\gamma_{1}}\times C_{\gamma_{2}}\) we denote the product weighted space with the norm
$$\bigl\Vert (u,v) \bigr\Vert _{\mathcal {C}}=\|u\|_{C_{\gamma_{1}}}+\|v \|_{C_{\gamma_{2}}}. $$

Definition 2.1

A function \(T:\Omega\times{\Bbb {R}}\to {\Bbb {R}}\) is called jointly measurable if \(T(\cdot,u)\) is measurable for all \(u\in{\Bbb {R}}\) and \(T(w,\cdot)\) is continuous for all \(w\in\Omega\).

Definition 2.2

A function \(f:I\times{\Bbb {R}}\times\Omega\to {\Bbb {R}}\) is called random Carathéodory if the following conditions are satisfied:
  1. (i)

    The map \((t,w)\to f(t,u,w)\) is jointly measurable for all \(u\in{\Bbb {R}}\), and

     
  2. (ii)

    the map \(u\to f(t,u,w)\) is continuous for all \(t\in I\) and \(w\in\Omega\).

     

Let E be a Banach space and \(T:\Omega\times E\to E\) be a mapping. Then T is called a random operator if \(T(w,u)\) is measurable in w for all \(u\in E\) and it expressed as \(T(w)u=T(w,u)\). In this case we also say that \(T(w)\) is a random operator on E. A random operator \(T(w)\) on E is called continuous (resp. compact, totally bounded and completely continuous) if \(T(w,u)\) is continuous (resp. compact, totally bounded and completely continuous) in u for all \(w\in\Omega\). The details of completely continuous random operators in Banach spaces and their properties appear in Itoh [14].

Definition 2.3

Let \({\mathcal {P}}(Y)\) be the family of all nonempty subsets of Y and C be a mapping from ω into \({\mathcal {P}}(Y)\). A mapping \(T:\{(w,y):w\in\Omega, y\in C(w)\}\to Y\) is called random operator with stochastic domain C if C is measurable (i.e., for all closed \(A\subset Y\), \(\{w\in\Omega, C(w)\cap A\neq\emptyset\}\) is measurable) and for all open \(D \subset Y\) and all \(y\in Y\), \(\{w\in\Omega: y\in C(w), T(w,y)\in D\}\) is measurable. T will be called continuous if every \(T(w)\) is continuous. For a random operator T, a mapping \(y:\Omega\to Y\) is called random (stochastic) fixed point of T if for P-almost all \(w\in\Omega\), \(y(w)\in C(w)\) and \(T(w)y(w)=y(w)\) and for all open \(D\subset Y\), \(\{w\in\Omega: y(w)\in D\}\) is measurable.

Now, we give some results and properties of fractional calculus.

Definition 2.4

([5, 18, 22])

The left-sided mixed Riemann–Liouville integral of order \(r>0\) of a function \(w\in L^{1}(I)\) is defined by
$$\bigl( I_{0}^{r}w \bigr) (t) =\frac{1}{\Gamma(r)} \int_{0}^{t}( t-s) ^{r-1}w(s)\,ds;\quad \mbox{for a.e. }t\in I, $$
where \(\Gamma(\cdot)\) is the (Euler’s) Gamma function defined by
$$\Gamma(\xi)= \int_{0}^{\infty}t^{\xi-1}e^{-t}\,dt; \quad \xi>0. $$
Notice that, for all \(r,r_{1},r_{2}>0\) and each \(w\in C\), we have \(I_{0}^{r}w\in C\), and
$$\bigl(I_{0}^{r_{1}}I_{0}^{r_{2}}w \bigr) (t)= \bigl(I_{0}^{r_{1}+r_{2}}w \bigr) (t);\quad \mbox{for a.e. }t\in I. $$

Definition 2.5

([5, 18, 22])

The Riemann–Liouville fractional derivative of order \(r\in(0,1]\) of a function \(w\in L^{1}(I)\) is defined by
$$\begin{aligned} \bigl(D^{r}_{0} w \bigr) (t) =& \biggl( \frac{d}{dt}I_{0}^{1-r}w \biggr) (t) \\ =&\frac{1}{\Gamma(1-r)}\frac{d}{dt} \int_{0}^{t}(t-s)^{-r}w(s)\,ds;\quad \mbox{for a.e. } t\in I. \end{aligned}$$
Let \(r\in(0,1]\), \(\gamma\in[0,1)\) and \(w\in C_{1-\gamma}(I)\). Then the following expression leads to the left-inverse operator.
$$\bigl(D_{0}^{r}I_{0}^{r}w \bigr) (t)=w(t);\quad \mbox{for all }t\in(0,T]. $$
Moreover, if \(I_{0}^{1-r}w\in C^{1}_{1-\gamma}(I)\), then the following composition is proved in [22]:
$$\bigl(I_{0}^{r}D_{0}^{r}w \bigr) (t)=w(t)-\frac{(I_{0}^{1-r}w)(0^{+})}{\Gamma(r)}t^{r-1};\quad \mbox{for all }t\in(0,T]. $$

Definition 2.6

([5, 18, 22])

The Caputo fractional derivative of order \(r\in(0,1]\) of a function \(w\in L^{1}(I)\) is defined by
$$\begin{aligned} \bigl({}^{c}D^{r}_{0}w \bigr) (t) =& \biggl(I_{0}^{1-r}\frac{d}{dt}w \biggr) (t) \\ =&\frac{1}{\Gamma(1-r)} \int_{0}^{t}(t-s)^{-r} \frac{d}{ds}w(s)\,ds; \quad \mbox{for a.e. }t\in I. \end{aligned}$$

In [13], Hilfer studied applications of a generalized fractional operator having the Riemann–Liouville and the Caputo derivatives as specific cases (see also [16, 24]).

Definition 2.7

(Hilfer derivative)

Let \(\alpha\in(0,1)\), \(\beta\in[0,1]\), \(w\in L^{1}(I)\), \(I_{0}^{(1-\alpha )(1-\beta)}\in AC^{1}(I)\). The Hilfer fractional derivative of order α and type β of w is defined as
$$ \bigl(D_{0}^{\alpha,\beta}w \bigr) (t)= \biggl(I_{0}^{\beta(1-\alpha)}\frac{d}{dt} I_{0}^{(1-\alpha)(1-\beta)}w \biggr) (t); \quad \mbox{for a.e. }t\in I. $$
(3)

Properties

Let \(\alpha\in(0,1)\), \(\beta\in[0,1]\), \(\gamma =\alpha +\beta-\alpha\beta\), and \(w\in L^{1}(I)\).
  1. 1.
    The operator \((D_{0}^{\alpha,\beta}w)(t)\) can be written as
    $$\bigl(D_{0}^{\alpha,\beta}w \bigr) (t)= \biggl(I_{0}^{\beta(1-\alpha)} \frac{d}{dt} I_{0}^{1-\gamma}w \biggr) (t) = \bigl(I_{0}^{\beta(1-\alpha)} D_{0}^{\gamma}w \bigr) (t); \quad \mbox{for a.e. } t\in I. $$
    Moreover, the parameter γ satisfies
    $$\gamma\in(0,1],\qquad \gamma\geq\alpha,\qquad \gamma>\beta,\qquad 1-\gamma < 1- \beta (1-\alpha). $$
     
  2. 2.
    The generalization (3) for \(\beta=0\), coincides with the Riemann–Liouville derivative and for \(\beta=1\) with the Caputo derivative.
    $$D_{0}^{\alpha,0}=D_{0}^{\alpha}\quad \mbox{and} \quad D_{0}^{\alpha,1}= {}^{c}D_{0}^{\alpha}. $$
     
  3. 3.
    If \(D_{0}^{\beta(1-\alpha)}w\) exists and is in \(L^{1}(I)\), then
    $$\bigl(D_{0}^{\alpha,\beta}I_{0}^{\alpha}w \bigr) (t)= \bigl(I_{0}^{\beta(1-\alpha )}D_{0}^{\beta (1-\alpha)}w \bigr) (t); \quad \mbox{for a.e. }t\in I. $$
    Furthermore, if \(w\in C_{\gamma}(I)\) and \(I_{0}^{1-\beta(1-\alpha )}w\in C^{1}_{\gamma}(I)\), then
    $$\bigl(D_{0}^{\alpha,\beta}I_{0}^{\alpha}w \bigr) (t)=w(t); \quad \mbox{for a.e. }t\in I. $$
     
  4. 4.
    If \(D_{0}^{\gamma}w\) exists and is in \(L^{1}(I)\), then
    $$\bigl(I_{0}^{\alpha}D_{0}^{\alpha,\beta}w \bigr) (t)= \bigl(I_{0}^{\gamma}D_{0}^{\gamma}w \bigr) (t) =w(t)-\frac{I_{0}^{1-\gamma}(0^{+})}{\Gamma(\gamma)}t^{\gamma-1};\quad \mbox{for a.e. }t\in I. $$
     

Corollary 2.8

Let\(h\in C_{\gamma}(I)\). Then the Cauchy problem
$$\textstyle\begin{cases} (D_{0}^{\alpha,\beta}u)(t)=h(t);\quad t\in I, \\ (I_{0}^{1-\gamma}u)(t)|_{t=0}=\phi, \end{cases} $$
has a unique solution given by
$$u(t)=\frac{\phi}{\Gamma(\gamma)}t^{\gamma-1}+ \bigl(I_{0}^{\alpha}h \bigr) (t). $$

From the above corollary, we conclude with the following lemma.

Lemma 2.9

Let\(f_{i}:I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\rightarrow{\Bbb {R}}\); \(i=1,2\)be such that\(f(\cdot,u_{1}(\cdot,w),u_{2}(\cdot,w),w)\in C_{\gamma_{i}}\)for all\(w\in \Omega\), and any\(u_{i}(w)\in C_{\gamma_{i}}\). Then the coupled system (1) is equivalent to the problem of solutions of the following system of fractional integral equations:
$$u_{i}(t,w)=\frac{\phi_{i}(w)}{\Gamma(\gamma_{i})}t^{\gamma _{i}-1}+ \bigl(I_{0}^{\alpha_{i}} f_{i} \bigl(\cdot,u_{1}(\cdot,w),u_{2}( \cdot,w),w \bigr) \bigr) (t);\quad w\in\Omega, i=1,2. $$

We need the following Itoh random fixed point theorem; see [14].

Theorem 2.10

LetXbe a nonempty, closed convex bounded subset of the separable Banach spaceEand let\(N:\Omega\times X\to X\)be a compact and continuous random operator. Then the random equation\(N(w)u=u\)has a random solution.

3 Coupled systems of Hilfer fractional random differential equations

In this section, we are concerned with the existence of solutions for the system (1). Let us start by defining what we mean by a random solution of the system (1).

Definition 3.1

By a random solution of the problem (1) we mean a coupled measurable function \((u_{1},u_{2}):\Omega\to C_{\gamma_{1}}\times C_{\gamma_{2}}\) that satisfies the conditions \((I_{0}^{1-\gamma_{i}}u_{i})(0^{+},w)=\phi_{i}(w)\); \(i=1,2\), and the equations \((D_{0}^{\alpha_{i},\beta_{i}}u_{i})(t,w)=f_{i}(t,u_{1}(t,w),u_{2}(t,w),w)\); \(i=1,2\) on \(I\times\Omega\).

The following hypotheses will be used in the sequel.
(H1)

The functions \(f_{i}\); \(i=1,2\) are random Carathéodory on \(I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\),

(H2)
there exist measurable and bounded functions \(p_{i},q_{i}:\Omega\to L^{\infty}(I,[0,\infty))\); \(i=1,2\), such that
$$\begin{aligned}& \bigl\vert f_{i}(t,u_{1},u_{2},w) \bigr\vert \leq\frac {p_{i}(t,w)|u_{1}|+q_{i}(t,w)|u_{2}|}{1+|u_{1}|+|u_{2}|}; \\& \quad \textit{for a.e. }t\in I, \textit{and each } u_{i}\in{\Bbb {R}}, w\in\Omega. \end{aligned}$$

Now, we shall prove the following theorem concerning the existence of random solutions of the system (1).

Theorem 3.2

Assume that the hypotheses (H1) and (H2) hold. Then the system (1) has at least one random solution defined on\(I\times\Omega\).

Proof

Define the following operators \(N_{i}:\Omega\times C_{\gamma _{i}}\rightarrow C_{\gamma_{i}}\); \(i=1,2\):
$$ \bigl(N_{i}(w)u_{i} \bigr) (t)= \frac{\phi_{i}(w)}{\Gamma(\gamma_{i})}t^{\gamma_{i}-1} + \int_{0}^{t}(t-s)^{\alpha_{i}-1}\frac {f_{i}(s,u_{1}(s,w),u_{2}(s,w),w)}{\Gamma (\alpha_{i})} \,ds, $$
(4)
and consider the continuous operator \(N:\Omega\times{\mathcal {C}}\to {\mathcal {C}}\) defined by
$$ N(w) (u_{1},u_{2})= \bigl(N_{1}(w)u_{1},N_{2}(w)u_{2} \bigr). $$
(5)
Set
$$p_{i}^{*}= \sup_{w\in\Omega} \bigl\Vert p_{i}(w) \bigr\Vert _{L^{\infty}}, \qquad q_{i}^{*}= \sup_{w\in \Omega} \bigl\Vert q_{i}(w) \bigr\Vert _{L^{\infty}}, \quad \mbox{and}\quad \phi_{i}^{*}= \sup _{w\in\Omega } \bigl\vert \phi _{i}(w) \bigr\vert ;\quad i=1,2. $$

For each \(i=1,2\), the map \(\phi_{i}\) is measurable for all \(w\in\Omega\). Again, as the indefinite integral is continuous on I, then \(N_{i}(w)\) defines a mapping \(N_{i}:\Omega\times C_{\gamma_{i}}\to C_{\gamma_{i}}\). Thus \((u_{1},u_{2})\) is a random solution for the system (1) if and only if \((u_{1},u_{2})=N(w)(u_{1},u_{2})\).

Next, for any \(u_{i}\in C_{\gamma_{i}}\); \(i=1,2\), and each \(t\in I\) and \(w\in \omega\), we have
$$\begin{aligned} \bigl\vert t^{1-\gamma_{i}} \bigl(N_{i}(w)u_{i} \bigr) (t) \bigr\vert \leq&\frac{|\phi_{i}(w)|}{\Gamma (\gamma _{i})}+\frac{t^{1-\gamma_{I}}}{\Gamma(\alpha_{i})} \int_{0}^{t}(t-s)^{\alpha_{i}-1} \bigl\vert f_{i} \bigl(s,u_{1}(s,w),u_{2}(s,w),w \bigr) \bigr\vert \,ds \\ \leq&\frac{|\phi_{i}(w)|}{\Gamma(\gamma_{i})}+\frac{t^{1-\gamma _{i}}}{\Gamma (\alpha_{i})} \int_{0}^{t}(t-s)^{\alpha_{i}-1} \bigl(p_{i}(s,w)+q_{i}(s,w) \bigr)\,ds \\ \leq&\frac{\phi_{I}^{*}}{\Gamma(\gamma_{i})}+\frac {(p_{i}^{*}+q_{i}^{*})T^{1-\gamma_{i}}}{\Gamma(\alpha_{i})} \int_{0}^{t}(t-s)^{\alpha_{i}-1}\,ds \\ \leq&\frac{\phi_{i}^{*}}{\Gamma(\gamma_{i})}+\frac {(p_{i}^{*}+q_{i}^{*})T^{1-\gamma_{i}+\alpha_{i}}}{\Gamma(1+\alpha_{i})}. \end{aligned}$$
Thus
$$ \bigl\Vert N(w) (u_{1},u_{2}) \bigr\Vert _{\mathcal {C}}\leq \sum_{i=1}^{2} \frac{\phi _{i}^{*}}{\Gamma (\gamma_{i})} +\frac{(p_{i}^{*}+q_{i}^{*})T^{1-\gamma_{i}+\alpha_{i}}}{\Gamma(1+\alpha_{i})}:=R. $$
(6)
This proves that \(N(w)\) transforms the ball
$$B_{R}:=B(0,R)= \bigl\{ (u_{1},u_{2})\in{\mathcal {C}}: \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal {C}} \leq R \bigr\} $$
into itself. We shall show that the operator \(N:\Omega\times B_{R}\to B_{R}\) satisfies all the assumptions of Theorem 2.10. The proof will be given in several steps.

Step 1. \(N(w)\) is a random operator on \(\Omega\times B_{R}\) into \(B_{R}\).

Since for each \(i=1,2\), \(f_{i}(t,u_{1},u_{2},w)\) is random Carathéodory, the map \(w\to f_{i}(t,u_{1},u_{2},w)\) is measurable in view of Definition 2.1. Similarly, the product \((t-s)^{\alpha_{i}-1}f_{i}(s,u_{1}(s,w), u_{2}(s,w),w)\) of a continuous and a measurable function is again measurable. Further, the integral is a limit of a finite sum of measurable functions, therefore, the map
$$w\mapsto\frac{\phi_{i}(w)}{\Gamma(\gamma_{i})}t^{\gamma_{i}-1} + \int_{0}^{t}\frac{(t-s)^{\alpha_{i}-1}}{\Gamma(\alpha _{i})}f_{i} \bigl(s,u_{1}(s,w),u_{2}(s,w),w \bigr)\,ds, $$
is measurable. As a result, \(N(w)\) is a random operator on \(\Omega\times B_{R}\) into \(B_{R}\).

Step 2. \(N(w)\)is continuous.

Let \(\{(u_{1n},u_{2n})\}_{n\in {\mathbb {N}}}\) be a sequence such that \((u_{1n},u_{2n})\rightarrow(u_{1},u_{2})\) in \(B_{R}\). Then, for each \(i=1,2\), \(t\in I\), and \(w\in\Omega\), we have
$$\begin{aligned}& \bigl\vert t^{1-\gamma} \bigl(N_{i}(w)u_{in} \bigr) (t)-t^{1-\gamma_{i}} \bigl(N_{i}(w)u_{i} \bigr) (t) \bigr\vert \\& \quad \leq\frac{t^{1-\gamma_{i}}}{\Gamma(\alpha_{i})} \int _{0}^{t}(t-s)^{\alpha _{i}-1} \bigl\vert f_{i} \bigl(s,u_{1n}(s,w),u_{2n}(s,w),w \bigr)-f \bigl(s,u_{1}(s,w),u_{2}(s,w),w \bigr) \bigr\vert \,ds. \end{aligned}$$
(7)
Since \((u_{1n},u_{2n})\rightarrow(u_{1},u_{2})\) as \(n\rightarrow \infty \) and \(f_{i}\) is random Carathéodory, then, by the Lebesgue dominated convergence theorem, Eq. (7) implies
$$\bigl\| N(w) (u_{1n},u_{2n})-N(w) (u_{1},u_{2}) \bigr\| _{\mathcal {C}} \to0\quad \text{as } n\to\infty. $$

Step 3. \(N(w)B_{R}\)is uniformly bounded.

This is clear since \(N(w)B_{R}\subset B_{R}\) and \(B_{R}\) is bounded.

Step 4. \(N(w)B_{R}\)is equicontinuous.

Let \(t_{1},t_{2}\in I\), \(t_{1}< t_{2}\) and let \((u_{1},u_{2})\in B_{R}\). Then, for each \(i=1,2\), and \(w\in\Omega\), we have
$$\begin{aligned} & \bigl\vert t_{2}^{1-\gamma_{i}} \bigl(N_{i}(w)u_{i} \bigr) (t_{2})-t_{1}^{1-\gamma_{i}} \bigl(N_{i}(w)u_{i} \bigr) (t_{1}) \bigr\vert \\ &\quad \leq \biggl\vert t_{2}^{1-\gamma_{i}} \int_{0}^{t_{2}}(t_{2}-s)^{\alpha_{i}-1} \frac {f_{i}(s,u_{1}(s,w),u_{2}(s,w),w)}{\Gamma(\alpha_{i})}\,ds \\ &\qquad {}-t_{1}^{1-\gamma_{i}} \int_{0}^{t_{1}}(t_{1}-s)^{\alpha_{i}-1} \frac {f_{i}(s,u_{1}(s,w),u_{2}(s,w),w)}{\Gamma(\alpha_{i})}\,ds \biggr\vert \\ &\quad \leq t_{2}^{1-\gamma_{i}} \int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha_{i}-1} \frac {|f_{i}(s,u_{1}(s,w),u_{2}(s,w),w)|}{\Gamma(\alpha_{i})}\,ds \\ &\qquad {} + \int_{0}^{t_{1}} \bigl\vert t_{2}^{1-\gamma_{i}}(t_{2}-s)^{\alpha_{i}-1} -t_{1}^{1-\gamma_{i}}(t_{1}-s)^{\alpha_{i}-1} \bigr\vert \frac {|f_{i}(s,u_{1}(s,w),u_{2}(s,w),w)|}{\Gamma(\alpha_{i})}\,ds \\ &\quad \leq t_{2}^{1-\gamma_{i}} \int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha_{i}-1} \frac {p_{i}(s,w)+q_{i}(s,w)}{\Gamma(\alpha_{i})}\,ds \\ &\qquad {} + \int_{0}^{t_{1}} \bigl\vert t_{2}^{1-\gamma_{i}}(t_{2}-s)^{\alpha_{i}-1} -t_{1}^{1-\gamma_{i}}(t_{1}-s)^{\alpha_{i}-1} \bigr\vert \frac {p_{i}(s,w)+q_{i}(s,w)}{\Gamma (\alpha_{i})}\,ds. \end{aligned}$$
Thus, we get
$$\begin{aligned} & \bigl\vert t_{2}^{1-\gamma_{i}} \bigl(N_{i}(w)u_{i} \bigr) (t_{2})-t_{1}^{1-\gamma_{i}} \bigl(N_{i}(w)u_{i} \bigr) (t_{1}) \bigr\vert \\ &\quad \leq\frac{(p_{i}^{*}+q_{i}^{*})T^{1-\gamma_{i}+\alpha_{i}}}{\Gamma (1+\alpha _{i})}(t_{2}-t_{1})^{\alpha_{i}} +\frac{p_{i}^{*}+q_{i}^{*}}{\Gamma(\alpha_{i})} \int _{0}^{t_{1}} \bigl\vert t_{2}^{1-\gamma _{i}}(t_{2}-s)^{\alpha_{i}-1} -t_{1}^{1-\gamma_{i}}(t_{1}-s)^{\alpha_{i}-1} \bigr\vert \,ds. \end{aligned}$$
As \(t_{1}\longrightarrow t_{2}\), the right-hand side of the above inequality tends to zero.

As a consequence of steps 1 to 4 together with the Arzelá–Ascoli theorem, we can conclude that \(N:\Omega\times B_{R}\to B_{R}\) is continuous and compact. Theorem 2.10 implies that the operator equation \(N(w)(u_{1},u_{2})=(u_{1},u_{2})\) has a random solution. This shows that the random system (1) has a random solution. □

4 Hilfer–Hadamard fractional random differential equations

Now, we are concerned with some existence results for the coupled system (2).

Set \(C:=C([1,T])\). Denote the weighted space of continuous functions defined by
$$C_{\gamma,\ln} \bigl([1,T] \bigr)= \bigl\{ w(t): (\ln t)^{1-\gamma}w(t)\in C \bigr\} , $$
with the norm
$$\|w\|_{C_{\gamma,\ln}}:= \sup_{t\in[1,T]} \bigl\vert (\ln t)^{1-r}w(t) \bigr\vert . $$

Let us recall some definitions and properties of Hadamard fractional integration and differentiation. We refer to [18] for a more detailed analysis.

Definition 4.1

(Hadamard fractional integral [18])

The Hadamard fractional integral of order \(q>0\) for a function \(g\in L^{1}([1,T])\), is defined as
$$\bigl({}^{H}I_{1}^{q}g \bigr) (x)= \frac{1}{\Gamma(q)} \int_{1}^{x} \biggl(\ln\frac {x}{s} \biggr)^{q-1}\frac{g(s)}{s}\,ds, $$
provided the integral exists.

Example 4.2

Let \(0< q<1\). Then
$${}^{H}I_{1}^{q} \ln t=\frac{1}{\Gamma(2+q)}(\ln t)^{1+q}, \quad \mbox{for a.e. }t\in[0,e]. $$
Set
$$\delta=x\frac{d}{dx},\qquad q>0,\qquad n=[q]+1, $$
and
$$AC_{\delta}^{n} := \bigl\{ u:[1,T]\to{\Bbb {R}} : \delta^{n-1} \bigl[u(x) \bigr]\in AC(I) \bigr\} . $$
Analogous to the Riemann–Liouville fractional calculus, the Hadamard fractional derivative is defined in terms of the Hadamard fractional integral in the following way.

Definition 4.3

(Hadamard fractional derivative [18])

The Hadamard fractional derivative of order \(q>0\) applied to the function \(w\in AC_{\delta}^{n}\) is defined as
$$\bigl({}^{H}D_{1}^{q}w \bigr) (x)= \delta^{n} \bigl({}^{H}I_{1}^{n-q}w \bigr) (x). $$
In particular, if \(q\in(0,1]\), then
$$\bigl({}^{H}D_{1}^{q}w \bigr) (x)=\delta \bigl({}^{H}I_{1}^{1-q}w \bigr) (x). $$

Example 4.4

Let \(0< q<1\). Then
$${}^{H}D_{1}^{q} \ln t=\frac{1}{\Gamma(2-q)}(\ln t)^{1-q},\quad \mbox{for a.e. }t\in[0,e]. $$
It has been proved (see e.g. Kilbas [17, Theorem 4.8]) that in the space \(L^{1}(I,{\Bbb {R}})\), the Hadamard fractional derivative is the left-inverse operator to the Hadamard fractional integral, i.e.
$$\bigl({}^{H}D_{1}^{q} \bigr) \bigl({}^{H}I_{1}^{q}w \bigr) (x)=w(x). $$
From Theorem 2.3 of [18], we have
$$\bigl({}^{H}I_{1}^{q} \bigr) \bigl({}^{H}D_{1}^{q}w \bigr) (x)=w(x)- \frac {({}^{H}I_{1}^{1-q}w)(1)}{\Gamma(q)}(\ln x)^{q-1}. $$

Analogous to the Hadamard fractional calculus, the Caputo–Hadamard fractional derivative is defined in the following way.

Definition 4.5

(Caputo–Hadamard fractional derivative)

The Caputo–Hadamard fractional derivative of order \(q>0\) applied to the function \(w\in AC_{\delta}^{n}\) is defined as
$$\bigl({}^{Hc}D_{1}^{q}w \bigr) (x)= \bigl({}^{H}I_{1}^{n-q}\delta^{n}w \bigr) (x). $$
In particular, if \(q\in(0,1]\), then
$$\bigl({}^{Hc}D_{1}^{q}w \bigr) (x)= \bigl({}^{H}I_{1}^{1-q}\delta w \bigr) (x). $$

From the Hadamard fractional integral, the Hilfer–Hadamard fractional derivative (introduced for the first time in [20]) is defined in the following way.

Definition 4.6

(Hilfer–Hadamard fractional derivative)

Let \(\alpha\in(0,1)\), \(\beta\in[0,1]\), \(\gamma=\alpha+\beta-\alpha \beta\), \(w\in L^{1}(I)\), and \({}^{H}I_{1}^{(1-\alpha)(1-\beta)}w\in AC^{1}(I)\). The Hilfer–Hadamard fractional derivative of order α and type β applied to the function w is defined as
$$\begin{aligned} \bigl({}^{H}D_{1}^{\alpha,\beta}w \bigr) (t)&= \bigl({}^{H}I_{1}^{\beta(1-\alpha )} \bigl({}^{H}D_{1}^{\gamma}w \bigr) \bigr) (t) \\ &= \bigl({}^{H}I_{1}^{\beta(1-\alpha)}\delta \bigl({}^{H}I_{1}^{1-\gamma }w \bigr) \bigr) (t);\quad \mbox{for a.e. }t\in[1,T]. \end{aligned}$$
(8)
This new fractional derivative (4.6) may be viewed as interpolating the Hadamard fractional derivative and the Caputo–Hadamard fractional derivative. Indeed for \(\beta=0\) this derivative reduces to the Hadamard fractional derivative and when \(\beta=1\), we recover the Caputo–Hadamard fractional derivative.
$${}^{H}D_{1}^{\alpha,0}= {}^{H}D_{1}^{\alpha} \quad \mbox{and}\quad {}^{H}D_{1}^{\alpha,1}= {}^{Hc}D_{1}^{\alpha}. $$

From Theorem 21 in [21], we concluded the following lemma.

Lemma 4.7

Let\(g_{i}:I\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\rightarrow{\Bbb {R}}\); \(i=1,2\)be such that\(g_{i}(\cdot,u_{1}(\cdot,w),u_{2}(\cdot,w),w)\in C_{\gamma_{i},\ln }([1,T])\)for any\(u_{i}(\cdot,w)\in C_{\gamma_{i},\ln}([1,T])\). Then problem (2) is equivalent to the following system of fractional integral equations:
$$u_{i}(t,w)=\frac{\psi_{i}(w)}{\Gamma(\gamma_{i})}(\ln t)^{\gamma_{i}-1} + \bigl({}^{H}I_{1}^{\alpha_{i}}g_{i} \bigl( \cdot,u_{1}(\cdot,w),u_{2}(\cdot,w),w \bigr) \bigr) (t); \quad w\in \Omega; i=1,2. $$
Now we give (without proof) an existence result for the system (2). Let us introduce the following hypotheses.
(\(\mathrm{H}'_{1}\))

The functions \(g_{i}\); \(i=1,2\) are random Carathéodory on \([1,T]\times{\Bbb {R}}\times{\Bbb {R}}\times\Omega\),

(\(\mathrm{H}'_{2}\))
there exist measurable and bounded functions \(p_{i},q_{i}:\Omega\to L^{\infty}([1,T],[0,\infty))\), such that
$$\begin{aligned}& \bigl\vert g_{i}(t,u_{1},u_{2},w) \bigr\vert \leq\frac{p_{i}(t,w)|u_{1}|+q_{i}(t,w)|u_{2}|}{1+|u_{1}|+|u_{2}|}; \\& \quad \mbox{for a.e. }t\in[1,T], \mbox{and each }u_{i}\in{\Bbb {R}}, w \in\Omega. \end{aligned}$$

Theorem 4.8

Assume that the hypotheses (\(\mathrm{H}'_{1}\)) and (\(\mathrm{H}'_{2}\)) hold. Then the coupled system (2) has at least one random solution defined on\([1,T]\times\Omega\).

5 An example

Let \(\Omega=(-\infty,0)\) be equipped with the usual σ-algebra consisting of Lebesgue measurable subsets of \((-\infty,0)\). As an application of our results we consider the following system of random Hilfer fractional differential equations:
$$ \textstyle\begin{cases} (D_{0}^{\frac{1}{2},\frac{1}{2}}u)(t,w)=f(t,u(t,w),v(t,w),w);\quad t\in [0,1], \\ (D_{0}^{\frac{1}{2},\frac{1}{2}}v)(t,w)=g(t,u(t,w),v(t,w),w);\quad t\in [0,1], \\ (I_{0}^{\frac{1}{4}}u)(t,w)|_{t=0}=(I_{0}^{\frac {1}{4}}v)(t,w)|_{t=0}=\frac{1}{1+w^{2}}, \end{cases}\displaystyle \quad w\in \Omega, $$
(9)
where
$$\begin{aligned}& \textstyle\begin{cases} f(t,u,v,w)=\frac{ct^{\frac{-1}{4}}|u|\sin t}{64(1+\sqrt {t})(1+w^{2}+|u|+|v|)};\quad t\in(0,1], u,v\in{\Bbb {R}}, w\in\Omega, \\ f(0,u,v,w)=0; \quad u,v\in{\Bbb {R}}, w\in\Omega, \end{cases}\displaystyle \\& g(t,u,v,w)=\frac{ct^{\frac{1}{4}}|v|}{64(1+w^{2}+|u|+|v|)}; \quad t\in[0,1], u,v\in{\Bbb {R}}, w\in\Omega, \end{aligned}$$
and \(c=\frac{9\sqrt{\pi}}{16}\). Clearly, the functions f and g are random Carathéodory.
The hypothesis (H2) is satisfied with
$$\textstyle\begin{cases} p_{1}(t,w)=\frac{ct^{\frac{-1}{4}}|\sin t|}{64(1+\sqrt{t})(1+w^{2})};\quad t\in(0,1], w\in\Omega, \\ p_{1}(0,w)=0;\quad w\in\Omega, \end{cases} $$
and \(q_{1}(t,w)=p_{2}(t,w)=0\), \(q_{2}(t,w)=\frac{ct^{\frac{1}{4}}}{64}\); \(t\in(0,1]\), \(w\in\Omega\). Hence, Theorem 3.2 implies that the coupled system (9) has at least one random solution defined on \([0,1]\times\Omega\).

Notes

Availability of data and materials

Not applicable.

Authors’ contributions

Each of the authors, SA, MB and YZ contributed equally to each part of this work. All authors read and approved the final manuscript.

Funding

The work was supported by the National Natural Science Foundation of China (No. 11671339).

Competing interests

The authors declare that they have no competing interests.

References

  1. 1.
    Abbas, S., Benchohra, M., Graef, J.: Coupled systems of Hilfer fractional differential inclusions in Banach spaces. Commun. Pure Appl. Anal. 17(6), 2479–2493 (2018) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abbas, S., Benchohra, M., Henderson, J., Lazreg, J.E.: Measure of noncompactness and impulsive Hadamard fractional implicit differential equations in Banach spaces. Math. Eng. Sci. Aerosp. 8, 1–19 (2017) Google Scholar
  3. 3.
    Abbas, S., Benchohra, M., Lagreg, J.-E., Alsaedi, A., Zhou, Y.: Existence and Ulam stability for fractional differential equations of Hilfer–Hadamard type. Adv. Differ. Equ. 2017, 180 (2017) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abbas, S., Benchohra, M., Lazreg, J.E., Zhou, Y.: A survey on Hadamard and Hilfer fractional differential equations: analysis and stability. Chaos Solitons Fractals 102, 47–71 (2017) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012) CrossRefGoogle Scholar
  6. 6.
    Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2015) zbMATHGoogle Scholar
  7. 7.
    Ahmad, B., Alsaedi, A., Kirane, M.: Nonexistence results for the Cauchy problem of time fractional nonlinear systems of thermoelasticity. Math. Methods Appl. Sci. 40, 4272–4279 (2017) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Aljoudi, S., Ahmad, B., Nieto, J.J., Alsaedi, A.: A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions. Chaos Solitons Fractals 91, 39–46 (2016) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Aljoudi, S., Ahmad, B., Nieto, J.J., Alsaedi, A.: On coupled Hadamard type sequential fractional differential equations with variable coefficients and nonlocal integral boundary conditions. Filomat 31(19), 6041–6049 (2017) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bharucha-Reid, A.T.: Random Integral Equations. Academic Press, New York (1972) zbMATHGoogle Scholar
  11. 11.
    Furati, K.M., Kassim, M.D.: Non-existence of global solutions for a differential equation involving Hilfer fractional derivative. Electron. J. Differ. Equ. 2013, 235 (2013) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Furati, K.M., Kassim, M.D., Tatar, N.-E.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64, 1616–1626 (2012) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) CrossRefGoogle Scholar
  14. 14.
    Itoh, S.: Random fixed point theorems with applications to random differential equations in Banach spaces. J. Math. Anal. Appl. 67, 261–273 (1979) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos 22(4), 1250086 (2012) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kamocki, R., Obczńnski, C.: On fractional Cauchy-type problems containing Hilfer’s derivative. Electron. J. Qual. Theory Differ. Equ. 2016, 50 (2016) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191–1204 (2001) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) zbMATHGoogle Scholar
  19. 19.
    Ladde, G.S., Lakshmikantham, V.: Random Differential Inequalities. Academic Press, New York (1980) zbMATHGoogle Scholar
  20. 20.
    Qassim, M.D., Furati, K.M., Tatar, N.-E.: On a differential equation involving Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2012, Article ID 391062 (2012) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qassim, M.D., Tatar, N.-E.: Well-posedness and stability for a differential problem with Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2013, Article ID 605029 (2013) MathSciNetGoogle Scholar
  22. 22.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Amsterdam (1987). Engl. Trans. from the Russian zbMATHGoogle Scholar
  23. 23.
    Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg; Higher Education Press, Beijing (2010) CrossRefGoogle Scholar
  24. 24.
    Tomovski, Ž., Hilfer, R., Srivastava, H.M.: Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct. 21, 797–814 (2010) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tsokos, C.P., Padgett, W.J.: Random Integral Equations with Applications to Life Sciences and Engineering. Academic Press, New York (1974) zbMATHGoogle Scholar
  26. 26.
    Wang, J.R., Feckan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, 806–831 (2016) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, J.R., Feckan, M., Zhou, Y.: Center stable manifold for planar fractional damped equations. Appl. Math. Comput. 296, 257–269 (2017) MathSciNetGoogle Scholar
  28. 28.
    Wang, J.R., Zhang, Y.: Nonlocal initial value problems for differential equations with Hilfer fractional derivative. Appl. Math. Comput. 266, 850–859 (2015) MathSciNetGoogle Scholar
  29. 29.
    Zhou, Y.: Attractivity for fractional evolution equations with almost sectorial operators. Fract. Calc. Appl. Anal. 21(3), 786–800 (2018) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhou, Y., Ahmad, B., Alsaedi, A.: Existence of nonoscillatory solutions for fractional neutral differential equations. Appl. Math. Lett. 72, 70–74 (2017) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhou, Y., Shangerganesh, L., Manimaran, J., Debbouche, A.: A class of time-fractional reaction–diffusion equation with nonlocal boundary condition. Math. Methods Appl. Sci. 41, 2987–2999 (2018) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zhou, Y., Vijayakumar, V., Murugesu, R.: Controllability for fractional evolution inclusions without compactness. Evol. Equ. Control Theory 4, 507–524 (2015) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhou, Y., Zhang, L.: Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems. Comput. Math. Appl. 73, 1325–1345 (2017) MathSciNetCrossRefGoogle Scholar

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Laboratory of Mathematics, Geometry, Analysis, Control and ApplicationsTahar Moulay University of SaïdaSaïdaAlgeria
  2. 2.Laboratory of MathematicsDjillali Liabes University of Sidi Bel-AbbèsSidi Bel-AbbèsAlgeria
  3. 3.Faculty of Information TechnologyMacau University of Science and TechnologyMacauP.R. China
  4. 4.Faculty of Mathematics and Computational ScienceXiangtan UniversityXiangtanP.R. China

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